I'm tryinge to compare two different proofs of the De Rham theorem: In one hand the (very much) abstract one using Cech cohomology and sheaves techniques, on the other hand the proof in Lee (Smooth manifolds, p480-487) which is actually taken form Bredon. Is there a way to explicit the isomorphisms from the sheaf theoretic proof to end up with the "De Rham" isomorphism as described in Lee? I started doing that but got stuck pretty quickly. Is there a reference for that ? If not any help would be appreciated.
For example for degree 1 we have the isomorphisms $$H^1_{DR}(M) = \frac{H^0(M,Z^1)}{d_*H^0(M,\Omega^0)}\simeq H^1(M,Z^0)=H^1(M,\underline{\mathbb{R}})=H^1_{sing}(M,\mathbb{R})\simeq H^1_{sing}(M,\mathbb{R})^*$$ and the "De Rham" isomorphism $H^1_{DR}(M,\mathbb{R})\rightarrow H^1_{sing}(M,\mathbb{R})^*$ given by $$\omega\mapsto \{c\mapsto \int_c\omega\}.$$